can grow rather rapidly in time. In fact, Kiselev and Sverak are able to construct
a vorticity solution of the 2D Euler equations in a disk whose gradient grows double exponentially in time [1]}. In the periodic setting, an example of Zlatos
shows that the vorticity gradient can grow at least exponentially [2]}. Choi and Jeong
obtain linear in time growth for the vorticity gradient for certain smooth and compactly supported initial vorticity in \(\mathbb {R}^2\)  [3]}.
Classical solutions to the 3D Euler equations could develop finite-time singularities ([4]}, [5]}). Many more results in this direction can be found in a review paper by Drivas and Elgindi [6]}.
As a special consequence, perturbations governed by the Euler equations near the trivial solution
are generally not stable. How much dissipation does one really need in order to achieve the
stability? Adding the full Laplacian dissipation is certainly sufficient. As demonstrated by
Schonbek and others (see, e.g., [7]}, [8]}, [9]}, [10]}), solutions of the Navier-Stokes equations
\({\left\lbrace \begin{array}{ll}\partial _t u + (u\cdot \nabla ) u =- \nabla P + \mu \Delta u, \\\nabla \cdot u =0\end{array}\right.}\) 
